Difference between revisions of "Q-exponential e sub q"

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The $q$-exponential $e_q$ is defined by the formula
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The $q$-exponential $e_q$ is defined for $|z|<1$ by the formula
$$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k}.$$
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$$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k},$$
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where $(q;q)_k$ denotes the [[q-shifted factorial]]. Note that this function is different than the [[q-exponential e sub 1/q |$q$-exponential $e_{\frac{1}{q}}$]].
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
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[[Exponential e in terms of basic hypergeometric phi]]
<strong>Theorem:</strong> The following formula holds:
 
$$e_q(z)=\dfrac{1}{(z;q)_{\infty}},$$
 
where $e_q$ is the [[Q-exponential e|$q$-exponential $E$]] and $(z;q)_{\infty}$ denotes the [[q-Pochhammer symbol]].
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
{{:Q-Euler formula for e sub q}}
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[[Q-Euler formula for e sub q]]
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=References=
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* {{PaperReference|q-Special functions, a tutorial|1994|Tom H. Koornwinder|prev=findme|next=findme}}  
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[[Category:SpecialFunction]]

Latest revision as of 03:30, 21 December 2016

The $q$-exponential $e_q$ is defined for $|z|<1$ by the formula $$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k},$$ where $(q;q)_k$ denotes the q-shifted factorial. Note that this function is different than the $q$-exponential $e_{\frac{1}{q}}$.

Properties

Exponential e in terms of basic hypergeometric phi

Q-Euler formula for e sub q

References